Paper: Multiple Regression

*The author of this computation has been verified*

R Software Module: rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Fri, 12 Dec 2008 04:34:29 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/12/t12290819686q8c1hjbsmt5pm0.htm/, Retrieved Fri, 12 Dec 2008 12:39:50 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2008/Dec/12/t12290819686q8c1hjbsmt5pm0.htm/}, year = {2008}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

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No (this computation is public)

User-defined keywords:

Dataseries X:

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189917 0 184128 0 175335 0 179566 0 181140 0 177876 0 175041 0 169292 0 166070 0 166972 0 206348 0 215706 0 202108 0 195411 0 193111 0 195198 0 198770 0 194163 0 190420 0 189733 0 186029 0 191531 0 232571 0 243477 0 227247 0 217859 0 208679 0 213188 0 216234 0 213587 0 209465 0 204045 0 200237 0 203666 0 241476 0 260307 0 243324 0 244460 0 233575 0 237217 0 235243 0 230354 0 227184 0 221678 0 217142 0 219452 0 256446 0 265845 0 248624 0 241114 0 229245 0 231805 0 219277 1 219313 1 212610 1 214771 1 211142 1 211457 1 240048 1 240636 1 230580 1

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation y[t] = + 202652.434482759 -30248.5034482758d[t] -15820.3353639846M1[t] -21151.8711877395M2[t] -31107.0341379310M3[t] -29050.9970881226M4[t] -25613.0593486590M5[t] -30037.0222988506M6[t] -35501.3852490422M7[t] -39891.3481992337M8[t] -45020.9111494253M9[t] -43879.0740996169M10[t] -8466.63704980843M11[t] + 1349.76295019157t + e[t] Multiple Linear Regression - Ordinary Least Squares Variable Parameter S.D. T-STAT H0: parameter = 0 2-tail p-value 1-tail p-value (Intercept) 202652.434482759 3029.726511 66.888 0 0 d -30248.5034482758 2671.385692 -11.3232 0 0 M1 -15820.3353639846 3457.2559 -4.576 3.5e-05 1.7e-05 M2 -21151.8711877395 3634.104897 -5.8204 1e-06 0 M3 -31107.0341379310 3631.503165 -8.5659 0 0 M4 -29050.9970881226 3629.674239 -8.0037 0 0 M5 -25613.0593486590 3622.572842 -7.0704 0 0 M6 -30037.0222988506 3617.524626 -8.3032 0 0 M7 -35501.3852490422 3613.247549 -9.8253 0 0 M8 -39891.3481992337 3609.744354 -11.051 0 0 M9 -45020.9111494253 3607.017294 -12.4815 0 0 M10 -43879.0740996169 3605.068131 -12.1715 0 0 M11 -8466.63704980843 3603.898127 -2.3493 0.023063 0.011531 t 1349.76295019157 53.023635 25.4559 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.978622613270707 R-squared 0.957702219204788 Adjusted R-squared 0.946002833027388 F-TEST (value) 81.8591851472408 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5697.64648914148 Sum Squared Residuals 1525769249.21563 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 189917 188181.862068965 1735.13793103487 2 184128 184200.089195402 -72.0891954023468 3 175335 175594.689195402 -259.689195402361 4 179566 179000.489195402 565.510804597669 5 181140 183788.189885058 -2648.1898850575 6 177876 180713.989885057 -2837.98988505745 7 175041 176599.389885057 -1558.38988505747 8 169292 173559.189885057 -4267.18988505749 9 166070 169779.389885057 -3709.38988505747 10 166972 172270.989885058 -5298.98988505754 11 206348 209033.189885057 -2685.18988505749 12 215706 218849.589885057 -3143.58988505749 13 202108 204379.017471264 -2271.01747126447 14 195411 200397.244597701 -4986.24459770114 15 193111 191791.844597701 1319.15540229885 16 195198 195197.644597701 0.35540229884154 17 198770 199985.345287356 -1215.34528735632 18 194163 196911.145287356 -2748.14528735634 19 190420 192796.545287356 -2376.54528735634 20 189733 189756.345287356 -23.3452873563323 21 186029 185976.545287356 52.4547126436624 22 191531 188468.145287356 3062.85471264368 23 232571 225230.345287356 7340.65471264366 24 243477 235046.745287356 8430.25471264367 25 227247 220576.172873563 6670.8271264367 26 217859 216594.4 1264.60000000000 27 208679 207989 690.000000000008 28 213188 211394.8 1793.2 29 216234 216182.500689655 51.4993103448367 30 213587 213108.300689655 478.699310344813 31 209465 208993.700689655 471.299310344822 32 204045 205953.500689655 -1908.50068965517 33 200237 202173.700689655 -1936.70068965518 34 203666 204665.300689655 -999.300689655154 35 241476 241427.500689655 48.4993103448295 36 260307 251243.900689655 9063.09931034483 37 243324 236773.328275862 6550.67172413786 38 244460 232791.555402299 11668.4445977012 39 233575 224186.155402299 9388.84459770117 40 237217 227591.955402299 9625.04459770117 41 235243 232379.656091954 2863.34390804599 42 230354 229305.456091954 1048.54390804599 43 227184 225190.856091954 1993.14390804599 44 221678 222150.656091954 -472.656091954007 45 217142 218370.856091954 -1228.85609195401 46 219452 220862.456091954 -1410.45609195400 47 256446 257624.656091954 -1178.65609195400 48 265845 267441.056091954 -1596.05609195402 49 248624 252970.483678161 -4346.48367816098 50 241114 248988.710804598 -7874.71080459768 51 229245 240383.310804598 -11138.3108045977 52 231805 243789.110804598 -11984.1108045977 53 219277 218328.308045977 948.691954022999 54 219313 215254.108045977 4058.89195402298 55 212610 211139.508045977 1470.49195402299 56 214771 208099.308045977 6671.691954023 57 211142 204319.508045977 6822.491954023 58 211457 206811.108045977 4645.89195402301 59 240048 243573.308045977 -3525.308045977 60 240636 253389.708045977 -12753.708045977 61 230580 238919.135632184 -8339.13563218398 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0741628393056531 0.148325678611306 0.925837160694347 18 0.0270456675344805 0.0540913350689611 0.97295433246552 19 0.00877966076144358 0.0175593215228872 0.991220339238556 20 0.00985980244324555 0.0197196048864911 0.990140197556754 21 0.00693700521735548 0.0138740104347110 0.993062994782645 22 0.0170689762050329 0.0341379524100658 0.982931023794967 23 0.0310763270839892 0.0621526541679785 0.96892367291601 24 0.0496938077337676 0.0993876154675353 0.950306192266232 25 0.0297494734024237 0.0594989468048475 0.970250526597576 26 0.0174837194949050 0.0349674389898099 0.982516280505095 27 0.0125250710182885 0.025050142036577 0.987474928981712 28 0.00735195468163626 0.0147039093632725 0.992648045318364 29 0.0043208061933941 0.0086416123867882 0.995679193806606 30 0.00259648248130405 0.0051929649626081 0.997403517518696 31 0.00161121710922137 0.00322243421844275 0.998388782890779 32 0.00235630473217778 0.00471260946435556 0.997643695267822 33 0.0065144773960634 0.0130289547921268 0.993485522603937 34 0.0320687799119045 0.0641375598238091 0.967931220088095 35 0.174013728124399 0.348027456248798 0.8259862718756 36 0.140248565886327 0.280497131772653 0.859751434113673 37 0.147503194030574 0.295006388061147 0.852496805969426 38 0.202188917793527 0.404377835587054 0.797811082206473 39 0.152797845981110 0.305595691962219 0.84720215401889 40 0.106475304919608 0.212950609839216 0.893524695080392 41 0.0660994448837287 0.132198889767457 0.933900555116271 42 0.0386296562233489 0.0772593124466979 0.96137034377665 43 0.0185952055397119 0.0371904110794239 0.981404794460288 44 0.0178661196325805 0.035732239265161 0.98213388036742 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.142857142857143 NOK 5% type I error level 14 0.5 NOK 10% type I error level 20 0.714285714285714 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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